Stat Chapter 20

robust

A confidence interval or significance test is called __________ if the confidence interval or P-value does not change very much when the conditions for use of the procedure are violated.

Do not reject H0 and say there is insufficient evidence to suggest that a difference exists between the mean GPAs. The rather large P-value suggests that the observed mean difference is not significant, and that there is no significant difference in GPA between twins in different living environments.

A group of psychologists was interested in knowing whether living environment had any effect on a student's grade point average (GPA). They took a set of twins and randomly assigned one sibling of the twins to live in an urban area and the other to live in a rural area After one year, they computed the GPAs for the twins and looked at the differences. Then they calculated the test statistic, and the P-value was found to be between 0.20 and 0.25. What can you conclude if the level of significance is 0.05?

fail to reject The rather large P-value suggests that the observed mean difference is not significant, and that there is no real difference in GPA between twins in different living environments.

A group of psychologists was interested in knowing whether living environment had any effect on a student's grade point average (GPA). They took a set of twins and randomly assigned one sibling of the twins to live in an urban area and the other to live in a rural area After one year, they computed the GPAs for the twins and looked at the differences. Then they calculated the test statistic, and the P-value was found to be between 0.20 and 0.25. With a 0.05 level of significance, we should ______ ____ _______ H0, and conclude that there is insufficient evidence to suggest that a difference exists between the mean GPAs of the two living environments.

twins. These siblings shared nearly identical genetic codes, but each was exposed to a different living environment (the treatment).

A group of psychologists was interested in knowing whether living environment had any effect on a student's grade point average (GPA). They took a set of twins and randomly assigned one sibling of the twins to live in an urban area and the other to live in a rural area. After one year, they computed the GPAs for the twins and looked at the differences. The "pairs" in this matched pairs design were the _________.

False The customers believe that the tires last less than 50,000 miles, so the alternative hypothesis should be μ < 50,000. The customers wouldn't be angry if the tires lasted 70,000 miles (and that's not equal to 50,000).

A tire manufacturer claims that one particular type of tire will last at least 50,000 miles. A group of angry customers believe that the tires do not last that long. They take a sample of 14 tires and want to test if the mean mileage of the tires is really 50,000. True or False: The customers want to test H0: μ = 50,000 versus Ha: μ ≠ 50,000.

Reject H0, and conclude that the tires were not performing as claimed. 0.02 is less than 0.05, so the p-value must be smaller than α . The customers have sufficient evidence to support their claim which is the alternative hypothesis.

A tire manufacturer claims that one particular type of tire will last at least 50,000 miles. A group of angry customers do not believe this is so and want to test if the mean mileage of the tires is really 50,000. After sampling and computing the test statistic, the customers discover 0.01 < P-value < 0.02. What decision should be made if testing at α = 0.05 level of significance?

Yes

Can a density curve be a t-distribution if its symmetric and centered at zero?

random sample

Do we have a random sample? If not, is the sample representative of the population? If not a representative sample, was it a randomized experiment?

the one-sample t test of significance

Draw an SRS of size n from a large population that has the Normal distribution with mean μ and standard deviation σ.

central limit theorem

Except in the case of small samples, the condition that the data are an SRS from the population of interest is more important than the condition that the population distribution is Normal because of the ___________ ________ ____________.

increases

For larger sample sizes, the test statistic ____________; resulting in a smaller p-value.

the population is from a Normal distribution

If the population is not from a Normal Distribution, then the sample size must be "large enough" with a shape similar to the Normal Distribution; then we apply the Central Limit Theorem.

large enough population: sample ratio

Is the population of interest ≥ 20 times 'n'?

It is unlikely that the population standard deviation σ will be known and the population mean μ will not be known.

It is unlikely that the population standard deviation σ will be known and the population mean μ will not be known. Copy.

dependent

Matched pairs is based upon _______________ samples... the data between the first and second lists are related.

Note that we have changed the "large enough sample" condition to be adaptable to the situations that we encounter. This is because t procedures are robust against violations of Normality.

Note that we have changed the "large enough sample" condition to be adaptable to the situations that we encounter. This is because t procedures are robust against violations of Normality. Copy.

Normal It is enough that the sample data be symmetric and single-peaked, but the population values are assumed to be Normally distributed.

One of the conditions for inference about a population mean states that the population values have a ___________ distribution.

observed effect

One way to demonstrate that a treatment causes an _____________ _________ is to use a matched pairs design for our experiment.

reject

P-value > 0.05 means we fail to _________ the null hypothesis.

20

Recall the conditions of having a random sample and having the population being at least _____ times the sample size.

matched The arms of the individuals represent the subjects in the matched pairs. The left and right arm of each individual are paired together with each arm receiving only one of the two formulas.

Researchers at a pharmaceutical company were developing a new formula for their sunscreen. They wanted to see if the new formula provided better protection against sunburns than the formula that was already on the market. They applied the new formula to one arm of an individual and the old formula to the other arm, randomly choosing the arms for each formula. They then compared the color difference between the arms. This study represents an example of a __________ design.

matched pairs design

Subjects are matched in pairs and each treatment is given to one subject in each pair. OR Observations are taken on the same subject before-and-after some treatment.

tdf will approach z

Suppose that for a particular family of T-distributions that t2 has 2 degrees of freedom (df) and t9 has 9 degrees of freedom. What would you anticipate happening to tdf as the df increase?

Larger test statistic; smaller p-value.

Suppose that for a right-tailed test of significance for a population mean with unknown standard deviation the sample mean exceeds the hypothesized mean and the sample standard deviation remains fixed. What happens to the test statistic and p-values as the sample size increases?

n = 10 The presence of the outlier prevents us from going any further with the t procedures when the sample size is small, which is the case in this problem since n = 10.

Suppose that we use a simple random sample of size n to estimate the mean commute time of students from their residence to campus. The histogram of the n commute times looks roughly symmetric but has one extremely large outlier. The robustness of our procedures will allow us to conduct inference in all of following cases except which one?

wider The spread of the t distributions is greater than that of the standard Normal distribution, so t* will always be greater than z* for any sample size. However, t* gets closer to z* as n increases.

Suppose we wish to calculate two level C confidence intervals for a population mean using the same sample mean and the same standard error for the sample mean. However, one CI will use the critical value from a t distribution, and the other will use the critical value from the standard Normal distribution. The CI using t* will be ________ than the CI using z*.

round down

Table C: Be conservative. When the exact df is not listed, "_________ _______" and use the closest df that does not exceed the df that is desired.

wider

Table C: Confidence intervals based upon "t" will be slightly _________ than those based upon "z."

z*

Table C: The final row includes 1.645, 1.96, & 2.576. These are "common confidence levels" & _____.

decrease

Table C: The t-distribution critical values ___________ as the degrees of freedom (df) increase.

H0: μ = 9.7 versus Ha: μ ≠ 9.7 Correct. The two-sided alternative hypothesis here reflects the concern that the fish might be too short or too long.

The mean length a particular species of fish should be 9.7 inches. The supervisor of a fish hatchery measures the length of a simple random sample of the fish every week after hatching to track their growth. If their average length is too short or too long, they are not maturing properly and further tests must be done to determine the cause. She randomly selects 30 fish and measures their individual lengths in inches to determine if they have grown to an average length of 9.7 inches. She observes an average length of 10.1 inches among the sample. What set of hypotheses is she interested in testing?

Do not reject H0, and conclude that the fish are at the proper length, on average. The P-value for 1.823 with 29 degrees of freedom is 0.0786 if using technology (p-value is between 0.05 and 0.1 if using Table C), it must be greater than α . Therefore we cannot reject H0 That means there is not enough evidence in the data that supports the fish are too long or too short.

The mean length a particular species of fish should be 9.7 inches. The supervisor of a fish hatchery measures the length of a simple random sample of the fish every week after hatching to track their growth. If their average length is too short or too long, they are not maturing properly and further tests must be done to determine the cause. She randomly selects 30 fish to determine if they have grown to a mean length of 9.7 inches. If the supervisor calculates a test statistic of 1.823, what should she conclude at the α = 0.05 level of significance?

means

The one-sample t-confidence interval is used to estimate ________.

standard error

The result when the standard deviation of a statistic is estimated from data. Of the sample mean x̄, it is s/√n.

one-sample z statistic

The standardized sample mean.

She is 95% confident that the true mean length of this species after five weeks of life is between 8.5 and 11.9 inches. For this sample the population of interest is the fish that have matured for five weeks.

The supervisor of a fish hatchery needs to measure the length of a particular species every week after hatching to track their growth. She randomly selects 100 fish from among those hatched five weeks ago and measures their individual lengths in inches. A histogram displays these lengths to have a roughly Normal distribution. She calculates the 95% confidence interval to be 10.2 ± 1.7 inches. What can she say with this result?

B. The data are centered at 0 with standard deviation of 1. The condition is that the lengths of all fish from this species have a Normal distribution, but it does not have to be the standard Normal distribution.

The supervisor of a fish hatchery needs to measure the length of a particular species every week to track their growth and verify that they maintain the mean length of that species at their age. She randomly selects 100 fish and measures their individual lengths in inches. To be able to conduct inference about the mean length, she hopes to see each of the following features in a histogram of the data except for which one? A. The distribution of the lengths is symmetric. B. The data are centered at 0 with standard deviation of 1. C. The histogram has just one peak. D. There are few to zero outliers.

normal

The t-distribution is not quite ___________.

skewness, outliers

The t-procedures guard against non-Normality except when there is strong _____________ or ____________ present.

responses

To compare the responses to the two treatments in a matched pairs design, find the difference between the ______________ within each pair. Then apply the one-sample t procedures to these differences

matched pairs t procedures

To compare the responses to the two treatments in a matched pairs design, find the difference between the responses within each pair. Then apply the one-sample t procedures to these differences.

robust In everyday language, robust can also mean strong, sturdy, or able to withstand poor conditions.

To say that our inference procedures are __________ means that the confidence level or P-value does not change very much when certain conditions of inference are violated.

The population must have a Normal distribution. Real-world data have a lot of variation, so no population or sample is expected to be perfectly Normal. Provided the sample size is large enough, we can still conduct inference procedures when the population is not Normal.

To say that the t procedures are robust means that we can still conduct inference in certain situations even if which of the following conditions is violated?

one-sample t-statistic

To test the hypothesis H0: μ = μ0, compute the _____-__________ ___-___________

False. The decision should be to not reject H0 and to say there is insufficient evidence to suggest that a difference exists between the mean GPAs. The rather large P-value suggests that the observed mean difference is not significant, and that there is no real difference in GPA between twins in different living environments.

True or False: A group of psychologists was interested in knowing whether living environment had any effect on a student's grade point average (GPA). They took a set of twins and randomly assigned one sibling of the twins to live in an urban area and the other to live in a rural area After one year, they computed the GPAs for the twins and looked at the differences. Then they calculated the test statistic, and the P-value was found to be between 0.20 and 0.25. With a 0.05 level of significance, we should reject H0, and conclude that a difference exists between the mean GPAs.

False More degrees of freedom correspond to more observations, which means that s does a better job of estimating σ. And that makes the t density curve approach the standard Normal curve. In other words, the t density curve approaches the standard Normal curve as its degrees of freedom increases.

True or False: As the degrees of freedom decrease, the t density curve approaches the standard Normal curve.

True. The test statistic is t = 1.826. There are 29 degrees of freedom, and the alternative hypothesis is two-sided.

True or False: The mean length a particular species of fish should be 9.7 inches. The supervisor of a fish hatchery measures the length of a simple random sample of the fish every week after hatching to track their growth. If their average length is too short or too long, they are not maturing properly and further tests must be done to determine the cause. She randomly selects 30 fish and observes a mean length of 10.1 inches and a standard deviation of 1.2 inches. The P-value for the corresponding test statistic must be between 0.05 and 0.10.

standard normal z-distribution

When standard deviation is known use the formula pictured. This statistic follows a ___________ ___________ ____-_____________.

t-distribution

When standard deviation is not known we can use s instead. This statistic is not quite Normal. It follows a ____-_____________.

40

When the data are not from a Normal distribution we also need to consider the following: Large enough sample: Are the data from a population that has a Normal distribution? If not and the sample size is at least ____, the t procedures can be used even for clearly skewed distributions.

15, 40

When the data are not from a Normal distribution we also need to consider the following: Large enough sample: Are the data from a population that has a Normal distribution? If not and the sample size is between ____ and ____, t procedures can be used except in the presences of outliers or strong skewness.

15

When the data are not from a Normal distribution we also need to consider the following: Large enough sample: Are the data from a population that has a Normal distribution? If not and the sample size is less than ____, t procedures can be used if the data close to Normal (roughly symmetric, single peak, no outliers)? If there is clear skewness or outliers then, do not use t.

standard error

When the standard deviation of a statistic is estimated from data, the result is called the ____________ ________ of the statistic. The standard error of the sample mean is pictured. Now the sample standard deviation will replace σ in many of our recent formulas, resulting in a new distribution.

μdiff, the population mean difference This is the unknown value we wish to estimate through inference.

When we choose to use a matched pairs design, it is because the parameter of interest is ________ in the responses between the two groups receiving the treatments.

one-sample t statistic

has the t-distribution with n - 1 degrees of freedom... With the sample standard deviation replacing σ; we can use this t-statistic for confidence intervals and tests of significance.

mean

population parameter

proportion of successes

population parameter

standard deviation

population parameter

mean

sample statistic and point estimate

proportion of successes

sample statistic and point estimate

standard deviation

sample statistic and point estimate

confidence interval for standard deviation

t* is the critical value for the t(n-1) density curve with area C between -t* and t*. This interval is exact when the population distribution is Normal and is approximately correct for large n in other cases.

standard deviation

x- is the best point estimate of population mean μ. Similarly, ___________ ___________ s can estimate standard deviation σ.